Ⅳ. Vector Fields on a Manifold 106
1. The Tangent Space at a Point of a Manilold 106
2. Vector Fields 115
3. One-Parameter and Local One-Parameter Groups Acting on a Manifold 122
4. The Existence Theorem for Ordinary Differential Equations 130
5. Some Examples of One-Parameter Groups Acting on a Manifold 138
6. One-Parameter Subgroups of Lie Groups 1- 45
7. The Lie Algebra of Vector Fields on a Manifold 1- 49
8. Frobenius Theorem 156
9. Homogeneous Spaces 164
Notes 171
Appendix Partial Proof of Theorem 4.1 172
Ⅴ. Tensors and Tensor Fields on Manifolds 175
1. Tangent Covectors 175
Covectors on Manifolds 176
Covector Fields and Mappings 178
2. Bilinear Forms. The Riemannian Metric 181
3. Riemannian Manifolds as Metric Spaccs 185
4. Partitions of Unity 191
Some Applications of the Partition of Unity 193
5. Tensor Fields 197
Tensors on a Vector Space 197
Tensor Fields 199
Mappings and Covariant Tensors 200
The Symmetrizing and Alternating Transformations 201
6. Multiplication of Tensors 204
Multiplication of Tensors on a Vector Space 205
Multiplication of Tensor Fields 206
Exterior Multiplication of A 207
ernating Tensors 207
The Exterior Algebra on Manifolds 211
7. Orientation of Manifolds and the Volume Element 213
8. Exterior Differentiation 217
An Application to Frobenius' Theorem 221
Notes 225
Ⅵ. Integration on Manifolds 227
1. Integration in Rn. Domains of Integration 227
Basic Properties of the Riemann Integral 228
2. A Generalization to Manifolds 233
Integration on Riemannian Manifolds 237
3. Integration on Lie Groups 241
4. Manifolds with Boundary 248
5. Stokes 's Theorem for Manifolds with Boundary 256
6. Homotopy of Mappings. The Fundnmental Group 263
Homotopy of Paths and Loops. The Fundamental Group 265
7. Some Applications of Differential Forms. The de Rham Groups 271
The Homotopy Operator 274
Ⅳ. Vector Fields on a Manifold 106
1. The Tangent Space at a Point of a Manifold 106
2. Vector Fields 115
3. One-Parameter and Local One-Parameter Groups Acting on a Manifold 122
4. The Existence Theorem for Ordinar Differemial Equations 130
5. Some Examples of One-Parameter Groups Actmg on a Manifold 13
6. One-Parameter Subgroups of Lic Groups 145
7. The Lie Algebra of Vector Fields on a Mamfold 149
8. Frobenius' Theorem 156
9. Homogeneous Spaces 164
Notes 171
Appendix Partial Proof of Theorem 41 172
Ⅴ. Tensors and Tensor Fields on Manifolds 175
1. Tangent Covectors 175
Covectors on Manifolds 176
Covector Fields and Mappings 178
2. Bilinear Forms. The Riemannian Metrie 181
3. Riemannian Manifolds as Metric Spaces 185
4. Partitions of Unity 191
Some Applications of the Partition of Unity 193
5. Tensor Fields 197
Tensors on a Vector Space 197
Tensor Fields 199
Mappings and Covariant Tensors 200
The Symmetrizing and Alternating Transformations 201
6. Multiplication of Tensors 204
Multiplication of Tensors on a Vector Space 205
Multiplication of Tensor Fields 206
Exterior Multiplication of Alternating Tensors 207
The Exterior Algebra on Manifolds 211
7. Orientation of Manifolds and the Volume Element 213
8. Exterior Differentiation 217
An Application to Frobenius' Theorem 221
Notes 225
Ⅵ. Integration on Manifolds 227
1. Integration in Rn. Domains of Integration 227
Basic Properties of the Riemann Integral 228
2. A Generalization to Manifolds 233
Integration on Riemannian Manifolds 237
3. Integration on Lie Groups 241
4. Manifolds with Boundary 248
5. Stokes 's Theorem for Manifolds with Boundary 256
6. Homnotopy of Mappings. The Fundamental Group 263
Homootopy of Paths and Loops. The Fundamental Group 265
7. Some Applications of Differential Forms. The de Rham Groups 271
The Homotopy Operator 274
8. Some Further Applications of de R ham Groups 278
The de Rham Groups of Lie Groups 282
9. Covering Spaces and the Fundamental Group 286
Notes 292
Ⅶ. Differentiation on Riemannian Manifolds 294
1. Differentiation of Vector Fields along Curves in Rn 294
The Geometry of Space Curves 297
Curvature of Plane Curves 301
2. Differentiation of Vector Fields on Submanifolds of Rn 303
Formulas for Covariant Derivatives 308
▽ x Y and Differentiation of Vector Fields 310
3. Differentiation on Riemannian Manifolds 313
Constant Vector.Fields and Parallel Displacement 319
4. Addenda to the Theory of Differentiation on a Manifold 321
The Curvature Tensor 321
The Riemannian Connection and Exterior Differential Forms 324
5. Geodesic Curves on Riemannian Manifolds 326
6. The Tangent Bundle and Exponential Mapping. Normal Coordinates 331
7. Some Further Properties of Geodesics 338
8. Symmetric Riemannian Manifolds 347
9. Some Examples 353
Notes 360
Ⅷ. Curvature 362
1. The Geometry of Surfaces in E3 362
The Principal Curvatures at a Point of a Surface 366
2. The Gaussian and Mean Curvatures of a Surface 370
The Theorema Egregium of Gauss 373
3. Basic Properties of the Riemann Curvature Tensor 378
4. The Curvature Forms and the Equations of Structure 385
5. Differentiation of Covariant Tensor Fields 391
6. Manifolds of Constant Curvature 399
Spaces of Positive Curvature 402
Spaces of Zero Curvature 404
Spaces of Constant Negative Curvature 405
Notes 410
REFERENCES 413
INDEX 417